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D..I APPLICATION NOTE 124

I'STITUTE OF TECHNOLOGY

HEALD E800 PEP"ER TREE LAN .S~NTA CLARA, CA 95051

True RMS Measurements

HEWLETTiWPACKARD

HEALDLIBRARY

True RMSMeasurements

June 5, 1970 '0 •....T- . ITUTE OF T-PEpoER TRE CH, ~OLOGY

u .rffA c ELANELARA, CA 95051

A

B

FIGURE 1.

Can you see any difference in the waveforms shown in Figure I? Both waveformswere computer generated. One waveform is distorted by 1% 3rd harmonic in phasewith the fundamental. The other waveform is a pure sine wave. 1% distortion is atypical specification for many good oscillators and is present in almost every sine waveused today. Would you be able to determine that waveform A in Figure 1 is distorted?Do you realize that if you measure the voltage level of this 1% distorted waveform withan average responding 4 digit DVM, only 3 digits are accurate, or with a 5 digit DVM,still only 3 digits are accurate? This AN will show that many measurements beingmade today are not "good enough".

1

EFFECTS ON MEASUREMENTS

The two major types of converters used with DVM's today are average respondingand true rms responding, The average responding converter responds to the averagevalue of the rectified waveform times a constant (constant ~ 1.11 for a pure sine wave),If the waveform contains harmonics of the fundamental, the average value will varydepending on the amplitude and phase of the harmonics, Multiplying this varyingaverage by a constant has to give a varying rms indication, Figure 2 is a plot of theerror caused by the common 1% distorted waveform in Figure 1. Each harmonic washeld to 1% of the fundamental and the phase of the harmonic varied from 0° to 360',

"'"',",

Q:oQ:Q:UJ

4I;

-...... V- J

J

'\ V:!

2 \ /--~ V~ .

i I- r----.- "\ \ i / V V 1--'

'>.--~,

r--::~~, ~~1---,

f-

~'- I

~~

~~~ /~ "If

1\'\~~ ~~VI I.-

1

\ "\i'-.--.-V V V./

~ """, ./V /2

\ VJ I\. /1

"-./iI

4 I

,d

O' 1800

PHASE

FIGURE 2. Percent Error versus Phase. Each harmonic is 1% ofthe fundamental magnitude.

Figure 3 is a plot of the % error in an average responding converter as a function ofthe mag'nitude of the distortion. This can be approximated by the simple equation:

,..., EllError = NEr

where Ell is the magnitude of the harmonic, Er is the magnitude of the fundamental,

and N is the order of the odd harmonic.Table 1 gives the equations for both average responding rms calibrated voltage

and true rms voltage. Note that only the average responding equation contains a phaseterm,

TABLE 1.

F(Il-A+ B sin ..... I + Csin(3wl+4»+ Zsin(nwl+e)

EAVG - +l to

[ A+ Bsin...... t + Csin (3 ,+ 4» + ....Jdl

E TRMS -~ jA2

+ B2

.

2

The 1% 3rd harmonic can add an additional ±O.33% error to the basic accuracyof an average responding converter. In a 4 digit DVM, this amounts to an additional33 counts of error, or the last digit is meaningless. In a 5 digit DVM, 333 counts oferror can be added, or the last 2 digits are meaningless. In both cases, only 3 digitsare meaningful with the typical 1% distorted waveform.

o dB(I DO!:)

-20 dBIIOZI

-60 dB -4{I dB

lo.m 1"'1HARMONIC DISTORTION

-80 dB

(D.Ol~."

0

/

3/ /

//V

i!/ ~ 3 - DIGIT

IDVM RESOLUTION

1

~~~$' o~ il

~ $:.~ aT- ....v ~.., ~ ~~~~ ..~.,,~ ~~Tf'.~ ..,

3,..."':V ...

i ,.:J./ ~ 4 - DIGIT~V" RESOLUTION

I /

/ V

3/ /

V/V,

t/./ ~ 5 - DIGIT DVM RESOLUTION

D.

.00-tOO dB(0.001 ~ )

~

>o

'"zozo11.::1 Q.a:

~a:

'"! .0

~~ .0

....Z

'"ua:'"a.. .00)(..~

FIGURE 3. Maximum % Error versus Harmonic Magnitude. Harmonic phase is constant.

So far, worst case has been assumed. This need not be done if the magnitude andphase of the harmonics can be determined. Determining the magnitude of the harmonicswould not be too difficult, but determining the phase relationship is next to impossible.It would be possible to analyze the signal using the wave analysis technique where eachharmonic is isolated and measured by an average responding converter. After the funda­mental and all the harmonics have been measured and recorded, they could be summedtogether by the square root of the sum of the squares method. This process eliminatesthe phase relationship problem, but requires expensive instrumentation and a great dealof time.

On the other hand, the true rms converter responds to the effective de heating valueof the waveform. The fundamental and each harmonic contribute to the total heatingvalue. The true rms converter in essence takes each component of the waveform andsums them by the square root of the sum of the squares technique. The converter doesthe job of the wave analysis and computing in a very economical way. Harmonicphase relationships have no effect on the response of the converter. The true rms con­version is a correct indication of the distorted waveform and is usable with 4 and 5digit DVM's.

The following example is used to exemplify the problems encountered with theaverage responding converter. The gain of an amplifier is determined by comparingthe relative input voltage E} to the relative output voltage E2 •

FIGURE 4. Amplifier Example

3

.9930

1.0043.9973

E1 contains the typical 1% harmonic distortion. Since it is difficult to determinewhich components are present, assume the worst case in which all of the distortion iscontained in the in-phase 3rd harmonic. The total error in an average responding con­verter will be the converter's inherent error plus the 33 counts of error due to theharmonic content of the waveform. A typical average converter specification is ±0.1 %,or the total worst case error of +43 counts for a 4 digit DVM.

If the amplifier adds no harmonic phase or amplitude distortion to the signal, the+43 counts will be present on the E2 voltage and the errors will cancel, but amplifiersare not pure. The amplifier, being a reactive device, will affect the phase relationshipof the harmonics to some extent. Since it is impossible to determine how much effectthe amplifier will have, worst case of 1800 phase shift of the harmonics must be assumed.The measurement of E2 will be in error by -23 counts (+10 counts inherent -33counts due to the harmonic-Refer to Figure 2). If in our example the gain of theamplifier is unity, the following can occur:

E1

E2

E2

E;This represents an error of 70 counts in the determination of the amplifier gain.

Your calculated amplifier gain is already 70 counts in error due to a phase distor­tion caused by the amplifier. If the amplifier adds 1% to the magnitude of the harmonics,the measurement of E2 will be in error by -56 counts (+10 counts inherent -66 countsdue to the distortion. Refer to Figure 3). The calculated gain is

E2 .9944~ = 1.0043 = .9901

or the calculated gain is in error by 89 counts.

By using a true rillS converter, you don't have to worry about the errors caused bydistortion. All of the digits remain useful and the reading is affected only by the basicaccuracy of the instrument. Distortion does not cause errors in true rillS converters.Distortion is part of the total power of the waveform and the converter responds to thepower, not the average of the waveform. Distortion would have to be less than 0.1 %to use all 4 digits of a 4 digit DVM and less than 0.01 % to use all 5 digits of a5 digit DVM. Signals just are not pure enough for you to gamble with average respond­ing measurements.

TRUE RMS MEASUREMENTS

Besides being the only accurate way to make everyday ac voltage measurements,the true rms converter allows you to accurately measure signals containing very highdistortion such as square waves, pulse trains, and even noise. This section discussessome of these unique measurement capabilities.

Square WaveA square wave can be generated by summing together sine waves of the appro­

priate frequency and amplitude. The harmonic content of a square wave is so greatthat the average responding converter calibrated to respond to the rms of a sine wavecannot even make an approximate measurement of the rms value. The true rms con­verter responds to the effective dc heating value of the square wave and is independentof the number or magnitude of the harmonics used to generate the square wave.

AC plus DCUntil now, all signals were ac coupled into a true rms converter. To measure the

rms value of an ac voltage plus a dc offset vOltage, you had to measure the ac componentand the dc component separately and then add them together as the square root of thesum of the square. Today, true rms converters are capable of accepting dc coupledsignals and measuring the rms value in one step. (Note Figure 5.)

4

·11.

'2 V

o Vde

9 V

I--- I - -

7.9 V

RMS '" JE <1(; 2 ~ E dc 2 ~~ 10.1

FIGURE 5. AC plus DC

Pulse Trains and Crest FactorA pulse train in essence is a non-symmetrical square wave. Only a true rms con­

verter can respond to a pulse train.

The term crest factor is used to describe a physical characteristic of any waveform.Basically crest factor is a ratio of the peak value of the signal to its rms value, or thesquare root of the inverse duty cycle minus one. Figure 6 shows both definitions.

CREST FACTORPOSITIVE VOLTAGE PEAK NEGATIVE VOLTAGE PEAK-----:R::-M"',S---- Of RMS

'NHICHEVER tS GREATER

CREST FACTOR~ Wh/:rc 0 ':" ~- {DUTY CYCLEI

FIGURE 6. Definition of Crest Factor

An average responding converter that can measure only pure sine waves accuratelyhas a crest factor of 1.414 (peak value of a sine wave to the rms value). True rmsconverters commonly have crest factors of 7: 1. The crest factor that a true rms con­verter can respond to is a function of the dynamic range of the input amplifier and theoperating point of the thermal converter. With a crest factor of 7: 1, you can measurethe rms value of a pulse train with a pulse width of 1/50 of the total period.

Also, to measure the rms value of a pulse that originates at 0 V dc, the signalmust be dc coupled into the true rms converter. This is a direct application of

jEuc2 + Edc

2•

5

NoiseNoise is a limiting factor in any system. Thermal noise, transistor noise, and

switch contact noise are examples of noise generated internally. This noise limits thesensitivity in measurement devices, distorts the purity of communication systems, andaffects the output signals of all amplifiers. To effectively reduce noise, you must beable to measure noise. Since noise level is referenced to power, true rms is the only wayto accurately measure the noise present in the system.

Mechanical to ElectricalStress, strain, vibration, shock, temperature expansion and contraction are mechani­

cal phenomena that are measured by converting mechanical motion into electricalsignals. These signals are often noisy, non-periodic, and usually non-sinusoida1. Inaddition, the signals are usually superimposed on some ac level, thus making true rmsthe only way to measure these signals accurately. More importantly, mechanical move­ment is a form of energy. To measure total mechanical energy, you convert to electricalenergy and measure the rms value of the electrical power. Only the thermal convertertechnique is a direct measurement of power.

THE USED-TO-BE'S

True rms measurements used to be slow. Today, 1 sec response time is possible.At frequencies below 20 Hz, measurement speed must be decreased to allow the con­verter time to respond to the signal. This is true for both average responding and truerms responding converters.

At high frequencies, true rms is faster than the majority of average respondingconverters. True rms can make accurate measurements at 1 reading a second on fre­quencies up to 1 MHz. Average responding converters are not capable of accurate 4digit measurements above 100 kHz.

True rms converters used to be expensive. They still cost more than averageconverters but how expensive is an incorrect measurement? When 4 or 5 digits ofaccuracy are required, it's less expensive to make a correct measurement the first timethan to correct a mistake caused by ·an inaccurate measurement.

SUMMARY

The true rms measurement technique is the most accurate available today. Besidesaccuracy, true rms allows a versatility in measurement capabilities unknown with pre­vious techniques. Hewlett Packard has realized the importance of true rms measure­ments and is making a definite commitment to the development of instrumentation in­corporating the advantages of true rms.

6

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